Non-singular plane curves with an element of “large” order in its automorphism group

Abstract

Let [Formula: see text] be the moduli space of smooth, genus [Formula: see text] curves over an algebraically closed field [Formula: see text] of zero characteristic. Denote by [Formula: see text] the subset of [Formula: see text] of curves [Formula: see text] such that [Formula: see text] (as a finite nontrivial group) is isomorphic to a subgroup of [Formula: see text] and let [Formula: see text] be the subset of curves [Formula: see text] such that [Formula: see text], where [Formula: see text] is the full automorphism group of [Formula: see text]. Now, for an integer [Formula: see text], let [Formula: see text] be the subset of [Formula: see text] representing smooth, genus [Formula: see text] curves that admit a non-singular plane model of degree [Formula: see text] (in this case, [Formula: see text]) and consider the sets [Formula: see text] and [Formula: see text]. In this paper we first determine, for an arbitrary but a fixed degree [Formula: see text], an algorithm to list the possible values [Formula: see text] for which [Formula: see text] is non-empty, where [Formula: see text] denotes the cyclic group of order [Formula: see text]. In particular, we prove that [Formula: see text] should divide one of the integers: [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Secondly, consider a curve [Formula: see text] with [Formula: see text] such that [Formula: see text] has an element of “very large” order, in the sense that this element is of order [Formula: see text], [Formula: see text], [Formula: see text] or [Formula: see text]. Then we investigate the groups [Formula: see text] for which [Formula: see text] and also we determine the locus [Formula: see text] in these situations. Moreover, we work with the same question when [Formula: see text] has an element of “large” order: [Formula: see text], [Formula: see text] or [Formula: see text] with [Formula: see text] an integer.